Worked Sample Problems - Math 114
CHAPTER 9 - Parametrized Curves and Polar Coordinates
Section 9.4, page 741
Problem 6
We eliminate the parameter as follows:
> | solve({x=4*sin(t),y=2*cos(t)},{x,t}); |
Or, of course , which is an ellipse. We'll plot the whole thing, and also the part that is covered by the parametrization as to goes from 0 to .
> | with(plots,display): |
> | A:=plot([4*sin(t),2*cos(t),t=-Pi..Pi],color=red,thickness=1): |
> | B:=plot([4*sin(t),2*cos(t),t=0..Pi],color=blue,thickness=3): |
> | display({A,B},scaling=constrained); |
The right half of the curve is traversed from top to bottom.
Problem 36
The point P rotates through the same angle as the wheel. So when the wheel has rotated clockwise through the angle , the center of the wheel has moved to the right. If the point P starts out at [0,a+b], then after the wheel has turned radians, it will be at [, ]. Let's plot this curve, with a=2 and b=1, a=2, b=2 (cycloid) and a=2, b=3 (spoke is longer than the radius of the wheel):
> | plot({seq([2*t+k*sin(t),2+k*cos(t),t=0..4*Pi],k=1..3)},color=blue,thickness=2,scaling=constrained); |
Section 9.5, page 749
Problem 9
> | x:=2*t^2+3; y:=t^4; |
Tangent line:
> | tanline:=subs(t=-1,y)+subs(t=-1,diff(y,t)/diff(x,t))*(xx-subs(t=-1,x)); |
(we used xx as the variable on the line because x was already taken!)
> | second_deriv:=diff(diff(y,t)/diff(x,t),t)/diff(x,t); |
Problem 18
> | x:=t^3; y:=3*t^2/2; |
> | arclength:=int(sqrt(diff(x,t)^2+diff(y,t)^2),t=0..sqrt(3)); |
Section 9.6, page 755
Problem 60
> | restart; |
> | simplify(subs(x=r*cos(theta),y=r*sin(theta),(x-5)^2+y^2=25)); |
> | solve(%,r); |
So the equation becomes .
Section 9.8, page 768
Problem 37
> | r:=1/(1+cos(theta)); |
> | plot([r*cos(theta),r*sin(theta),theta=0..2*Pi],view=[-4..4,-4..4],color=blue,thickness=2); |
Clearly this is a parabola, vertex is when , so directrix is the line when x=1.
Section 9.9, page 775
Problem 15
First plot, then solve for intersections and go from there:
> | r1:=6; r2:=3*csc(theta); |
> | plot({[r1*cos(theta),r1*sin(theta),theta=-Pi..Pi],[r2*cos(theta),r2*sin(theta),theta=-Pi/2..Pi/2]},color=blue,thickness=2,scaling=constrained,view=[-7..7,-7..7]); |
We want above the line and below the circle -- where do they intersect?
> | solve(r1=r2); |
Looks like and -- so we integrate:
> | area:=int(r1^2/2-r2^2/2,theta=Pi/6..5*Pi/6); |